Free arrangements and relation spaces

Discrete & Computational Geometry, Jul 1994

Yuzvinsky [7] has shown that free arrangements are formal. In this note we define a more general class of arrangements which we callk-formal, and we show that free arrangements arek-formal. We close with an example which distinguishesk-formal arrangements from formal arrangements.

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Free arrangements and relation spaces

Discrete Comput Geom Free Arrangements 0 1 Relation Spaces 0 1 0 2 Department of Mathematics, University of Wisconsin-Madison , Madison, WI 53706 , USA 1 1Department of Mathematics, University of Kansas , Lawrence, KS 66045 , USA Yuzvinsky [7] has shown that free arrangements are formal. In this note we define a more general class of arrangements which we call k-formal, and we show that free arrangements are k-formal. We close with an example which distinguishes k-formal arrangements from formal arrangements. Let  be a field a n d let V be a n / - d i m e n s i o n a l v e c t o r s p a c e o v e r K. A hyperplane in V is a c o d i m e n s i o n 1 s u b s p a c e o f V. A n arrangement ~t in V is a finite set of h y p e r p l a n e s . L e t {xl . . . . . xt} be a basis for t h e d u a l V* a n d let S be the s y m m e t r i c a l g e b r a of V* w h i c h is i s o m o r p h i c t o the p o l y n o m i a l a l g e b r a KI-xl . . . . . x J . T h e n e a c h h y p e r p l a n e H in I / h a s a defining f o r m - 1. Introduction orR = a l x I -'k "'" -k a/X/ with ker(gn) = H, u n i q u e u p to a c o n s t a n t multiple. T h u s a n a r r a n g e m e n t d c a n * The first author was supported in part by a U.S. Department of Education Fellowship. The second author was supported in part by the National Science Foundation. be described by the product of such forms. Given an arrangement d , let Q = Q ( ~ ) = I-I ct.. H E ~r Note that Q is unique up to constant multiple. The arrangement d is completely determined by Q. Let D(~r be the S-module which consists of all derivations 0: S --* S such that 0(Q) is a multiple of Q. We call D(~r the module of ~r When D(~r is a free S-module, d is said to be free. There are some interesting connections between the lattice L ( d ) of intersections of elements of d and the S-module D(d). For a survey, see the book by Orlik and Terao [ 3 ]. In this work we study these connections further. In particular, we are interested in the study of formal arrangements, introduced by Falk and Randell [ 1 ]. An arrangement d is formal if all linear dependencies among the defining forms of the hyperplanes of d are generated by dependencies corresponding to the codimension 2 subspaces in L(d). Using techniques developed in [ 6 ], Yuzvinsky has shown in 1-7] that free arrangements are formal. In this paper we introduce a more general class of arrangements called k-formal arrangements. By definition, d is 2-formal if d is formal. The main result of this paper is that if d is free, then d is k-formal for 2 < k < r - 1, where r is the codimension of the common intersection of all the hyperlanes of d . Our proof does not use techniques from [ 6 ]. This theorem imposes strong conditions on the ways in which the hyperplanes of a free arrangement intersect each other. In particular, we have some combinatorial inequalities for free arrangements. We close the paper with an example of an arrangement which is 2-formal (formal) but not 3-formal. This distinguishes k-formal arrangements from formal arrangements. 2. Preliminaries We begin with some definitions and a few basic facts which are used throughout this paper. For further background, see [-3]. Fix an arrangement d . Let n = I~r Let S and Q ( d ) be as described in the introduction. We call Q ( d ) the defining polynomial of ~r Definition 2.1. Let L = L(~r be the set of intersections of elements of ~ . Define a partial order on L by X ~ Y ~ Y ~ X . Note that this is reverse inclusion. Thus V is the unique minimal element. Definition 2.2. Define a rank function on L by r(X) = codim X. Thus r(V) = 0 and r(H) = 1 for H 9 ~ . Call H an atom of L. Let X, Y 9 L. Then their meet X ^ Y is equal to ('] { Z 9 u Y ~ Z}. Their join X v Yis equal to X n Y. [] Proof. See Proposition 4.8 of [-3]. Corollary 2.8. I f g~ ~ ~ , then D ( ~ ) ~_ D(~). Thus if X, Y 9 L ( ~ ) , with X __ Y, then D(~r) _ D(~x). Definition 2.9. An arrangement ~ is called a free arrangement if D ( ~ ) is a free module over S. Theorem 2.10. I f d is free, then ~ x is free for each X 9 L. 3. Formal Arrangements Fix an arrangement ~. Choose a defining form aH for each H ~ ~ . We wish to study the various relations among these forms. We consider the vector space E(~/) which has a basis consisting of the symbols {eH[H 6 ~/}. We have a map = 9 H e d Ke.v* given by eu ~ uu- Now define F ( ~ ) to be the kernel of this map. We call F(~/) the relation space and we refer to elements of F ( d ) as relations. Then we have an exact sequence Definition 3.1. An element w = ~ crier of E(~/) has length p if precisely p of the ca 6 K are nonzero. We write p = length(w). Definition 3.2. Let n = I ~ 1 . Let 2 _ p _< n - 1. We define Fp(~) to be the subspace of F(,~/) spanned by all relations of length no greater than p + 1: Fp(~l) = <{w = ~ cHeB ~ F(~/)llength(w) _< p + 1}). Definition 3.3. 2-generated. We say ~ is p-oenerated if F ( d ) = Fp(~/). Call ~ / f o r m a l if d is Remark 3.4. Let r = r(~). For r < p _< n - 1, Fp(~/) = F(~). Thus all arrangements are r-generated. Note that M formal is the strongest of the above conditions. Note that if r < 2, then it is always the case that F(M) = F2(~/). Thus, for the rest of the paper, we assume that r > 3. Remark 3.5. F o r each X ~ L, we have an inclusion F2(,~/x) ~ F(~). If ~ / i s formal, then each relation in F ( ~ ) can be written as a sum of relations in F2(,.Q/). Thus , d is formal if and only if we have a surjection ~2: ( ~ X e L r(X): 2 F2(,~x) -'~ F(~/) --' O. Note that if r(X) = 2, then F 2 ( d x ) = F(..q/x). Free Arrangementsand RelationSpaces 53 Example 3.6. Consider the real 3-arrangements ~r and ~r defined by Q1 = X 1 X 2 X 3 ( X 1 -- X2)(X2 -- X3)(X1 -- 2-X3) and Q2 = X 1 X 2 X 3 ( X I "~- X2 dl- X3), respectively. Then ~r is formal, whereas ~r is not. Remark 3.7. Although each linear dependency among the defining forms of the hyperplanes of ~r is associated to a particular subspace in L, it is not true that the property of being formal depends only on L (see Example 2.2 of [7"1). Note, however, that ~r formal does impose some conditions on L. By Remark 3.5, we have r, ( l ~ x l - 2) _> n - r ( ~ ' ) . XcL r~X) = 2 In [ 1 ] Falk and Randell asked if free arrangements are formal. This question was resolved by Yuzvinsky: Theorem3.8 [7, Corollary 2.5]. I f ~t is free, then ~ is formal. We reprove this by proving our main theorem in the next section. Using this theorem, we can give some easy lattice conditions which are necessary for an arrangement to be free. In addition to the inequality in Remark 3.7, we have the following proposition. The proof is straightforward and is left to the reader. Call H ~ ~r a separator if r(~r - {H}) < r(~r Proposition 3.9. Let ~ be an arrangement. Fix H ~ d and let ~ ' = ~ - {H}. (i) I f H is a separator, then ~ is formal if and only if ~t' is formal. (ii) Suppose H is not a separator, l f ~ / is formal, then H contains a rank 2 element of H ~ ' ) . Example 3.10. The real 4-arrangement ~r defined by ~2 = xlx2x3x4(Xx - x 2 ) ( x l + x3 - x4) is not free. Proof. The plane H = ker(xl + xa - x4) is not a separator, and does not contain any rank 2 subspace of L ( ~ r {H}). Thus by the previous proposition and Theorem 3.8, ~r is not free. Note, however, that H does contain the rank 3 subspace ker(xl) n ker(xa) c~ker(x4). [] 4. A Generalization of Formal Arrangements We now define the class of k-formal arrangements and give a generalization of Theorem 3.8. Suppose ~r is formal. Then we have an exact sequence ( ~ XeL r(X) = 2 0 -~ R 3 ( ~ ) ~ F2(~x) -~ F(~) ~ O, where Ra(0ff) is the kernel of the map n2. Thus Ra(~r ) is a space of relations among the relations corresponding to the rank 2 elements of L. Note that, for any X e L, we have an inclusion Ra(dx) ~ R3(~r ). If the map ( ~ XEL r(X) = 3 Rs(~/x) -~ Ra(~t ) is surjective, then we call ~ 3-formal. We can continue in this fashion to define k-formal for k > 3. We make this idea precise in the following definitions. Definition 4.1. Let R o ( ~ ) = T(~t)*. Remark 4.2. Note that, for each X E L, there is a restriction map Ro(~x) ~ Ro(~r Definition 4.3. For 1 _< k _< r, define Rk(d~ ) recursively to be the kernel of the map X~L r(X) = k - 1 Rk - l(.~x) ~ R k_ 1(.~), where the maps nk are defined as follows. For each k _> 1 and for each Y e L with r(Y)_> k, we have a map ik(Y): Rk(~r)~Rk(a~,), which can be viewed as inclusion. The map 7~k is a sum of these maps. Also, we have a commutative diagram: 0 0 ' Rk+l(~r) ' ( ~ Rk((~r)x) ~kt~ Rk(~y ) X<Y ' Rk+ l(a~r ' Rk(~x) ,,(d) Rk(~). Since X < Y, we have (Mr)x = Mx. The middle vertical map Jk(Y) is a direct sum of identity maps and zero maps, so it can be viewed as inclusion. N o t e that if r ( X ) = 0, then we have X = V. Thus R I ( M ) is the kernel of the restriction m a p V * ~ T(M)*, so R I ( M ) = T(M) ~ N o t e also that we have identifications R dMn) = T(Mn) ~ ~- H ~ ~- ~ e n. Hence R2(M ) can be identified with the kernel of the m a p ~ e n ~ T(M) ~ so we see that R2(M ) ~_ F(M). Definition 4.4. F o r 2 < k < r - 1, define the class of k-formal arrangements as follows: (i) An arrangement d is 2-formal if M is formal. (ii) F o r k > 3, M is k-formal if d is (k - 1)-formal and the m a p XeL r(X) = k is a surjection. O u r main theorem states that if d is free, then M is k-formal for 2 < k < r - 1. To prove this theorem, we first relate the vector spaces Rk(M) to the S-module of derivations D(M). We define S-modules Dk(M ) for 0 < k < r. O u r construction is analogous to that of the vector spaces Rk(M ). Definition 4.5. Let D o ( d ) = D(M). Recall that for each X e L, we have an inclusion m a p Do(M ) --, Do(Mx). D e n o t e this m a p by tpo(X). Definition 4.6. F o r 1 ~ k < r, define the S-modules Kk(d~[) and Dk(M) recursively to be the kernel and cokernel respectively o f the m a p rk-, = Tk- I(M): Dk- I(M) ~ Dk- t(Mx), @ XeL r(X) =k - 1 where the m a p Zk is a sum of maps ~k(Y): Dk(~C)--*Dk(Slr). F o r Y 6 L with r(Y) > k - 1, these m a p s ~pk(Y) are defined by the following diagram: ( ~ Xr v(X) = k - 1 ~ ) X<Y r(X) = k - 1 Dk_,(sr ,k-,(~) Dk_,(Mx ) , Dk(M ) I tPIt-I(Y) Pit- I(Y) r D k- 1(~r ~*-'(~'~ Dk-,((~Cr)x) , Dk(Mr) , 0 , 0 . R e m a r k 4.7. Note that Kt(~qr = 0 for any arrangement, since it is the kernel of the inclusion m a p Zo: Do(M) -+ Derx(S). Also K 2 ( , ~ ) = 0, for if [0] is in the kernel of the m a p z,:Dl(~r 0 H~,rr D~('ClH), then 0 e Do(~Cn) for each H e ~ . R e m a r k 4.8. If ~r is the empty arrangement, then D l ( d ) = 0, and hence Ok(,.r16)2 = 0 for 1 ~ k _< r. Remark 4.9. Note that D1(~r ) = Derx(S)/D(~). g: T ( ~ ) - , Do(~')o given by We have an isomorphism l g(v) = ~ /=1 x,(v)D, for each v e T(~r Thus DerK(S)o ~ V and D1(~r "~ V / T ( ~ ) . Proposition 4.10. For 0 < k < r, we have Dk(d)* ~--Rk!~). Proof. Recall that Ro(~r ) = T(~r By the r e m a r k above, we have an isomorphism fo(~r = g*: Do(d)* -* Ro(~t) for any arrangement ~ . For k ___O, consider the following diagram: 0 , R,+,(z~r "it, Rk(.~). r(X) = k ~ XEL , First note that the rows are exact. It is straightforward to show that this diagram commutes for k = 0, and hence we can construct an isomorphism f1(~r D I ( ~ ) * ~ R1(~r ), for any arrangement ~r We continue via induction, noting that it is sufficient to show that the following diagram commutes for each X e L with r(X) = k: Dk(dX), ~Cx~ Dk(d)~ I fk(~eIX) I fk(~ Rk(~/x) ,,(x) Rk(d). By chasing this diagram, we can show that, for each k < r, the diagram commutes and hence we have an isomorphism fk(~r Dk(~r ~ Rk(~t ). The idea is to combine the above diagram with those from Definitions 4.3 and 4.6. [] We may view the correspondence X ~ - ~ D ( X ) = D(~x) for X e L as a contravariant functor on L, because there is the inclusion m a p D(X) --, D(Y) whenever X > Y. Note that D(T(~r D(~r and D ( V ) = DerK(S). It turns out that this functor is local (in the sense of Definition 6.4 of [ 4 ]): Proposition 4.11. Let X e L and ga ~ Spec(S), and define N H. He~tx ~ue~ Then the localization at ga of the inclusion map D(X) ~ D(X(ga)) is an isomorphism. Proof. This is contained in Example 4.123 of [3]. [] L e m m a 4.12. For 0 <_k <_r, the functors DR(X) = Dk(dX) for X ~ L are local. Proof. This follows using induction and the fact that localization preserves exact sequences. [] The following result is essentially L e m m a (5.15) of 15], but since our setup is a bit different, we give a detailed p r o o f here. Proposition 4.13. I f s# is free, then, for 1 <~k < r, we have (i) pds D k - l ( d ) = k -- 1, (ii) ht go = k - 1 for each fJ e Asss Dk- l ( d ) , (iii) Kk(~r = O. Before the proof, let us recall a fundamental fact from commutative algebra. F o r details (in the local ring case) see P r o p o s i t i o n 3.9 in C h a p t e r VI of [2-]. Theorem 4.14. For any finitely oenerated oraded S-module M, depth s M < m i n { l - h t p } < $~9 Asss M m a x { l - h t g o } = ~oe Asss M max { l - h t g o } ~ e Supps M = dim s M. Here, depth s M is defined with respect to the maximal h o m o g e n e o u s ideal of S. P r o o f o f 4.13. We use induction. W h e n k = 1 we k n o w pds Do(M) = 0 since M is free, so (i) is clear. Also, since Do(M ) is free, the annihilator of each 0 ~ Do(M ) is the zero ideal. Thus Asss Do(M ) consists of the zero ideal, and now (ii) follows. In R e m a r k 4.7 we saw that K I ( M ) = 0 so (iii) holds. Suppose the result holds for some k (1 < k < r - 1). Then, since K k ( M ) = 0, we have an exact sequence XEL r(X) = k- 1 Dk_ l(Mx) -* Dk(M ) ---*O. T h u s pds Dk(.~[) <--k. N o w let ~o e Spec(S) with ht ~ _< k - 1. We write M~ = Mr(~,Xv ). T h e n we have the exact sequence o -, x(M )o 0 XeL r(X) = k - 1 Dk_ l(Mx{r))~ ~ Dk(M)to ~ O. N o t e that there is at m o s t one X e L of rank k - 1 (in the case when ht go = k - 1) with O k_ l(Jdx(~)) :f: 0. Thus either Dk- I(M~)~ ~ Dk- l(MX{r))~ ( ~ XEL r(X) =k - 1 is an i s o m o r p h i s m o r D k_ l(MX{~)) = 0 for all X e L with r(X) = k - 1. In either case, Dk(M)~ = 0. This shows that, for a n y go e Supps Ok(,~f), we have ht ~ >_ k, and hence, by T h e o r e m 4.14, we k n o w depth s Dk(M ) _< dim s Dk(M ) <_ I -- k. NOW by the A u s l a n d e r - B u c h s b a u m formula (for graded modules), we have l = pd s D~(M) + depth s Dk(M) <---pds Dk(M) + l -- k. We have seen that pd s Dk(M ) <_ k, so it follows that pds Dk(M) = k. This shows (i). Also, l -- k = depths D k ( d ) ~ dims Dk(M) <_ l -- k. Thus Dk(d~r is Cohen-Macaulay, and, by Theorem 4.14, each go e Asss Dk(,J~) has height k. This shows (ii). To show (iii), consider the sequence Dk(~X). Suppose there is some ~o e Asss Kk+ 1(~r Then, since Asss Kk+ 1 ( ~ ) -----Asss Dk(,~r we know that ht 9 = k by (ii). We localize to obtain the exact sequence 0 --* Kk+ 1(~)~ ~ Dk(~'~)~ ~ D~(.~xt~))~. ( ~ XeL r(X) = k Since ht go = k, there is at most one X e L with r(X) = k such that Dk(~X(~o))# 0, so either the m a p D,(~r Dk(~c4X(,o))~, ~ XeL r(X) = k is an isomorphism o r Dk(dp) = 0. This shows that Kk+ l ( ~ r = 0. This is impossible, however, since Asss Kk§ 1(~r -----Supps Kk+ l ( d ) [2, Proposition 2.9 in Chapter VI]. Thus it must be the case that A s S s K k + l ( d ) = ~ . This implies that K,+ 1(~r = 0 [2, Proposition 2.4 in Chapter VII. Hence (iii) is established and the proof is complete. [] Now we have our main theorem: Theorem 4.15. I f d is free, then d is k-formal for 2 < k < r - 1. Proof. Suppose 2 < k < r - 1. Then, by part (iii) of the previous proposition, we have an exact sequence of vector spaces o--, o (d)o --, 0 XEL r(X) = k Dk(dX) o --* Dk +l ( d ) o ~ O. Now, by Proposition 4.10, we know that Dk(d)J ~ Rk(~r ) for 0 < k < r. Once again we have a commutative diagram with the same maps as in the proof of Proposition 4.10. Recall that the maps in the second and third columns are isomorphisms. ~ D k + , ( d ) ~ Dk(dX)~ , D k ( d ) ~ s 0 ' Rk+,(,~r Rk(dX) ~k ' Rk(d). Now we see that the m a p nk is onto. Hence d is k-formal. [] 0 0 ~ ( ~ X~L r(X) = k , ( ~ X~L rlX) = k This theorem gives additional conditions on L for free arrangements. Suppose ~r is free. Then, by Theorem 2.10, Jzrr is free and hence k-formal for each Y e L , 0 < k < r(IO - 1. Consider Y e L with r(IO > 2. By Remark 3.7, we have Z X<Y r(X) = 2 ( l ~ x l - 2) _> I ~ r l - r(Y). F r o m the exact sequence we see that @ 0 ~ Ra(~r) ~ F2(~x) ~ F ( ~ r ) -~ 0, dim R3(Mr) = r(Y) - I~rl + ( l ~ x l - 2). X.<Y r(X) = 2 Thus, for example, since ~r is 3-formal, the surjectivity of Y~L r(r) = 3 yields the following: Corollary 4.16. I f ~ is free and r(~r > 4, then YcL r(r) = 3 (3-[dr[+ ~ ([~r X~Y r(X) = 2 ~ (1~r X~L r(X) = 2 There are similar inequalities corresponding to k-formal for k > 4. 5. AnExample We now distinguish k-formal arrangements from formal arrangements. Example 5.1. The real 4-arrangement ~/ described below is formal but not 3-formal. Free Arrangementsand Relation Spaces Let Q(~r = I~.l~ aj, where 61 ~1 = X3, OC2 ~ X 3 - - X4, OC3 ~ X2, O~4 = X 2 + X 3 - - 2 X 4 , ~5 = X l , ~6 = XI + X3 - - 2X4, a7 = x2 + 2x3 - 2x,, a8 = xl + 2x3 - 2x4, 69 = X1 + X 2 + X 3 - - 2x,, We computed L = L(~r by hand and later verified our calculations using a computer program written by John Keaty. Note that relations a m o n g the defining forms come from subspaces X ~ L such that r(X) < I~r Thus, in describing L, we do not include the so-called "generic" subspaces X ~ L where r(X) = I~xl. We use the following notation to describe the remaining elements of L. The intersection H~ n Hj is denoted by the corresponding indices ij. We do the same for intersections of three or more hyperplanes. Each lattice element is denoted by its maximal index set. F o r example 14 = 17 = 147, so 147 is listed below. R a n k 3 : 1 2 9 1 0 , 3 6 9 1 0 , 4 5 9 1 0 1 3 6 8 9 , 1 4 5 7 9 , 1 4 6 7 8 , 2 3 5 7 8 , 2 3 6 7 9 , 2 4 5 8 9 , 3 4 5 6 9 1 2 3 4 7 1 0 , 1 2 5 6 8 1 0 Rank 2: 1 2 1 0 , 1 4 7, 1 6 8 , 2 3 7 , 2 5 8 , 3 6 9 , 4 5 9 We first show that a~r is formal. We count seven nongeneric rank 2 elements, each an intersection of three hyperplanes. Thus we have seven relations of length 3. They are: This leads to the matrix ~1 - - ~ 2 - - ~10 ~ O, 2a2 + a5 - ~(s = O, ~t + ~ 4 - - ~ 7 = 0 , ~1 + ~ 6 - - ~ 8 = O, 2a2 + a s - - ~ 7 = O, o~4 --{- o~5 - o~9 = O. This matrix has r a n k 6, and thus dim F2(.~r ) --- 6. Recall that we have an exact sequence Since d is essential, dim T(~r ~ = 4, and thus dim F ( d ) = 6. Hence F2(,~ ) = F ( d ) and d is formal. N o w show that d is not 3-formal. Since ~r is formal, we have an exact sequence 0 ~ R3(.~ ) ~ (~) F2(,~/x) -~ F(,~a/) ~ 0. XEL r(X) = 2 N o t e that each (nongeneric) rank 2 element X e L is an intersection of three hyperplanes, so each F2(,.~x) will have dimension 1. Thus the middle term in the a b o v e sequence has dimension 7, and dim R 3 ( d ) = 1. In particular, R 3 ( ~ ) ~ 0. To show that the m a p is not surjective, we show that, for each Y e L with r(Y) = 3, we have R 3 ( d r ) = 0. This a m o u n t s to showing that, for each such Y, the m a p O YeL r(Y) = 3 R3(d1") "~ R 3 ( d ) ( ~ X e L (,~ly) r(X) = 2 FE((~r)x) -~ F(dr) is an injection. Each T(~Cr) ~ has dimension 3, so dim F(~Cr) = I~yl - 3. The chart below lists the nongeneric rank 3 elements Y e L, along with the nongeneric rank 2 elements of L(dv): Rank 3 element Contained by rank 2 element Difference 4 5 9 1 0 3 6 9 1 0 F o r each Y in the chart, the relations corresponding to the rank 2 elements of L(dr) are linearly independent. Thus R3(~/r) = 0 for each Y of rank 3 so ~r is not 3-formal. In particular, ~r is not free. F r o m this chart, we also see that the number of nongeneric rank 2 elements of L(~Ir) is always equal to I~r - 3. Thus each ~tr is formal, so that ~t is actually locally formal [7, Definition 2.3]. Note that we can show ~r is not free directly, by using Corollary 4.16 and the information in the chart above. Alternatively, ~r is not free by Yuzvinsky's inequality [6, Corollary 3.2] as well as by Terao's factorization theorem [3, Theorem 4.137]. 1. M. Falk and R. Randell , On the homotopy theory of arrangements , Adv. Stud. Pure Math . 8 ( 1986 ), 101 - 124 . 2. E. Kunz , Introduction to Commutative Algebra and Algebraic Geometry , Birkh~iuser, Basel, 1985 . 3. P. Orlik and H. Terao , Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften , Vol. 300 , Springer-Verlag, Berlin, 1992 . 4. L. Solomon and H. Terao , A formula for the characteristic polynomial of an arrangement, Adv . in Math. 64 ( 1987 ) 305 - 325 . 5. H. Terao , Generalized exponents of a free arrangement of hyperplanes and Shephard-ToddBrieskorn formula , Invent. Math . 63 ( 1981 ), 159 - 179 . 6. S. Yuzvinsky , Cohomology of local sheaves on arrangement lattices , Proc. Amer. Math. Soc . 112 ( 1991 ), 1207 - 1217 . 7. S. Yuzvinsky , First two obstructions to the freeness of arrangements , Trans. Amer. Math. Soc . 335 ( 1993 ), 231 - 244 .


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K. A. Brandt, H. Terao. Free arrangements and relation spaces, Discrete & Computational Geometry, 1994, 49-63, DOI: 10.1007/BF02574365